Collisions of localized patterns in a nonvariational Swift-Hohenberg equation

We first consider the variational problem with $\epsilon=0$ and employ numerical continuation to construct the bifurcation diagram shown in Fig. 1, cf. [13]. A trivial flat state $u=0$ exists at all $r$, and undergoes a subcritical bifurcation to a periodic pattern at $r=0$ (red profile in lower left inset in Fig. 1). Four snaking branches bifurcate from the periodic pattern branch in secondary bifurcations, corresponding to the symmetric LSs $L_{0}$, $L_{\pi}$ and the antisymmetric LSs $L_{\pi/2}$, $L_{3\pi/2}$ mentioned in the introduction. The branches overlap in pairs owing to the symmetry $u\to-u$, and both sets display characteristic snaking behavior in the vicinity of the Maxwell point $r_{M}$ [13, 9]. In addition, rung states connect the symmetric and antisymmetric snaking branches, arising in pitchfork bifurcations close to every saddle-node bifurcation on these branches [18]. Each rung actually corresponds to four branches of unstable asymmetric localized solutions, related by the symmetries $x\to-x$ and $u\to-u$ [14]. Examples of symmetric and antisymmetric solution profiles are shown in the lower left inset in Fig. 1, together with the periodic pattern state. When the LSs fill the available domain the snaking branches reconnect to the periodic pattern.

Figure 2 shows the free energy of the periodic pattern from Fig. 1 versus $r$. The stable part of the periodic pattern state (thick blue line) has a free energy which decreases monotonically with $r$ and changes sign at the Maxwell point $r_{M}\approx-0.675$.

Here we describe how the bifurcation structure changes in the nonvariational case, specifically for $\epsilon=0.03$ (Fig. 3). As in the case $\epsilon=0$, there is a trivial branch $u=0$, a periodic pattern branch emerging subcritically from it, and two snaking branches. However, the two snaking branches in Fig. 3 now correspond to $L_{0}$ and $L_{\pi}$, since these states are no longer related by symmetry, and consequently snake in phase before reconnecting to the periodic state. Moreover, the $L_{\pi/2}$, $L_{3\pi/2}$ states and the rung states reconnect, forming a sequence of ’Z’-shaped branches consisting of asymmetric solutions.

As a consequence of the nonvariational structure of the problem when $\epsilon>0$, these asymmetric solutions drift and hence may collide. The structure of several Z branch solutions at different $r$ values is shown in the inset in Fig. 3. It is important to observe that the Z branch states may be stable or unstable. Stable solutions are present on the “diagonal” part of each Z, i.e. in the range $-0.70\lesssim r\lesssim-0.63$ (except for the lowest branch, which is stable only between $-0.65\lesssim r\lesssim-0.62$). The corresponding drift velocity $c$ depends on $r$, as shown in Fig. 4 for several of the states depicted in Fig. 3. In Fig. 4, the stable part of any given Z branch corresponds to the segment between $-0.70\lesssim r\lesssim-0.63$, where the drift speed is largest. We emphasize that longer asymmetric LSs drift more slowly than short ones.

To compute the drift velocity of asymmetric LSs using perturbation theory in the limit of weak symmetry breaking, $0<\epsilon\ll 1$, we introduce the fast and slow time variables

and denote their spatial phase by $\theta(T)$. Following the calculation presented in [17], we posit the expansion

where $U_{0}$ is a known moving pattern whose propagation velocity we seek to determine. At leading order, Eq. (1) implies

while at $O(\epsilon^{\frac{1}{2}})$ we obtain

where the operator $\mathcal{L}\equiv r-(1+\partial_{x}^{2})^{2}+3b_{2}U_{0}^{2}-5U_{0}^{4}$ is self-adjoint.

Differentiating Eq. (6) with respect to $x$, one finds that $\mathcal{L}U_{0}^{\prime}=0$, i.e. if $U_{0}$ solves (6), then $U_{0}^{\prime}$ solves (7). In [17], the authors considered $r$ to be asymptotically close to the edge of the snaking interval, with a focus on the dynamics of depinning. This required taking into account two additional solutions which lie in the null space of $\mathcal{L}$: one symmetric and one asymmetric mode. Here, we instead consider asymmetric states which are located within the snakes-and-ladders structure, away from the saddle-nodes of the snaking branches. Consequently $U_{0}^{\prime}$ is the only function in the null space of $\mathcal{L}^{\dagger}=\mathcal{L}$. Since this translation mode is already included in the Ansatz (5) we can set $u_{1}\equiv 0$. Finally, at $O(\epsilon)$, we obtain

Multiplying by $U_{0}^{\prime}$ and integrating over the domain, using the fact that $\mathcal{L}$ is self-adjoint and that $\mathcal{L}U_{0}^{\prime}=0$, we obtain

Rearranging, we arrive at the drift velocity,

a special case of a more general result [22].

Equation (10) is invariant under $(\epsilon,U_{0})\to(-\epsilon,-U_{0})$, a symmetry that is inherited from Eq. (1). For symmetric profiles $U_{0}(x)$, the numerator vanishes; symmetric states therefore remain at rest even when $\epsilon>0$. However, for the asymmetric profiles shown in Fig. 3 the velocity $c$ is nonzero. This is due to the nonsinusoidal nature of these profiles and in particular the contribution from the fronts at either end of the LS profile. We define the number $n$ of significant extrema in any given LS as the number of extrema whose amplitude is larger than $1/e$ times the maximum amplitude in the LS, and the number of significant wavelengths as $1/2$ the number of significant extrema. The adjective significant will be dropped in the following when there is no ambiguity. For solutions with many significant wavelengths, $n\gg 1$, the numerator converges to a constant, while the denominator grows approximately linearly with $n$. This implies that $c\propto 1/n$ at large $n$, which is quantitatively consistent with the results from numerical continuation, as shown in Fig. 5: longer structures move more slowly, as already highlighted in the discussion of Fig. 4.

To determine the range of validity of the prediction in Eq. (10), we perform direct numerical simulations (DNSs) of Eq. (1) for selected values of the parameter $\epsilon$. We also perform numerical continuation in $\epsilon$. The results reported below extend those in [22] where the asymptotic result was compared with numerical continuation at only one value of $\epsilon$, $|\epsilon|=0.01$. For this purpose, we consider a one-dimensional periodic domain of the same length $L=40\pi$ as in the numerical continuation and solve Eq. (1) using a semi-implicit pseudo-spectral numerical scheme for spatial derivatives [23] and a fourth-order Runge-Kutta time-stepping scheme. All DNS results presented in this paper were obtained on a finely resolved uniform spatial grid of $8192$ grid points (corresponding to approximately $100$ grid points per pattern wavelength), except when specified otherwise. We use a time step of $dt=0.01$. Larger values of $dt$ led to incorrect drift speeds.

To verify Eq. (10), we consider a state on the second lowest Z branch at $\epsilon=0.13$, $r=-0.66$, shown in the inset in Fig. 6. The state is asymmetric: a small change in peak/trough amplitudes with increasing $\epsilon$ can be discerned, a consequence of symmetry breaking when $\epsilon>0$.

Figure 6 shows the drift velocity $c$ as a function of $\epsilon$, with excellent quantitative agreement between DNSs, numerical continuation and theory for $\epsilon\lesssim 0.1$. For $\epsilon\gtrsim 0.1$, the numerically obtained drift velocities deviate from the asymptotics, with the asymptotic prediction overestimating the numerical value by less than $10\%$. However, the DNSs and numerical continuation continue to show excellent agreement. At $\epsilon\approx 0.13$ the state under consideration ceases to exist, since beyond this point the bifurcation structure of the system is qualitatively altered (cf. [18]). Additional runs with the smaller time step $dt=0.001$ were also performed, and gave the same results as those for $dt=0.01$, indicating that the DNSs are well resolved in time. We also repeated the DNSs on a coarser grid with $2048$ grid points and obtained drift velocities $c$ that are indistinguishable from those shown in Fig. 6, indicating that the simulations are also well resolved in space.

Here we describe the rich collision phenomenology that is observed when different types of LSs collide at various values of $r$, as illustrated in Fig. 7. We distinguish the following four collision scenarios:

• •

  Scenario A: collision between two asymmetric states which differ in the number of significant wavelengths, and travel in the same direction, with the shorter, faster LS catching up to the longer, slower LS, as predicted by Eq. (10). The two colliding extrema are of opposite sign. We focus specifically on collisions between two LSs of length two and three wavelengths. Four different possible outcomes are observed: deletion of one extremum, formation of a drifting bound state without a change in the number of extrema, and the creation of one or four new extrema. Note that the bound states at $r=-0.68$ and $r=-0.645$ shown in Fig. 7 differ in their separation. The cases $r=-0.7$ and $r=-0.635$ in fact involve two separate consecutive collisions due to the periodicity of the domain, of which only the first is shown in Fig. 7. In the former case, the second collision (not shown) results in a drifting bound state, while in the latter case (shown in Fig. 8), the asymmetric LS is rendered symmetric and stationary owing to the nucleation of an additional extremum, resulting in a larger but stationary bound state.

• •

  Scenario B: a drifting asymmetric state collides with a stationary symmetric state with a maximum at its center. The two colliding extrema are of opposite sign. We focus on a two-wavelength asymmetric LS and a symmetric LS with three positive and two negative large-amplitude extrema. Four different possible collision outcomes are observed: deletion of one extremum, formation of a drifting bound state without change in the number of extrema, and the creation of one or four new extrema.

• •

  Scenario C: same as scenario B but with the symmetric LS flipped by $u\to-u$ so that the two colliding extrema are of the same sign. We focus on a two-wavelength asymmetric LS and a symmetric structure with three negative and two positive large-amplitude extrema. Five different possible collision outcomes are observed: deletion of one extremum, formation of a drifting bound state without change in the number of extrema, and the creation of one, three or five new extrema.

• •

  Scenario D: head-on collision between two asymmetric states. The two colliding extrema are of the same sign. We focus on a pair of identical two-wavelength patterns (collisions between asymmetric LSs of distinct sizes were also considered, and gave qualitatively similar results). Four different possible collision outcomes were observed: deletion of two extrema and the creation of three or five new extrema. No pure bound states were observed in this case.

Figure 7 shows a summary of the space-time plots depicting the different possible collision outcomes. All collisions were simulated using DNS on a uniform grid of $2048$ points to facilitate longer simulation times. Some cases were also repeated with $8192$ grid points and no change in the collision behavior was observed.

Three qualitatively distinct collision outcomes are observed across all collision scenarios: deletion of extrema, formation of a bound state, and the creation of new extrema. Some of these states travel while others are symmetric and hence stationary. One notices that scenarios A and B are quite similar to one another in terms of the outcome realized at a given $r$ (except near $r=-0.645$, see also Fig. 12). Scenarios C and D are similar in the same sense. Since the closest extrema in the collisions in scenarios A and B are both of opposite sign but are of the same (negative) sign in scenarios C and D, one might surmise that the relative sign of interacting extrema is important in determining the qualitative collision outcome, cf. [18]. Our detailed results broadly support this suggestion but indicate that this sign is not the sole factor determining the collision outcome since differences between scenarios persist.

Figure 7 reveals that the speed $c$ of the traveling state plays a significant role in determining the collision outcome. This speed is controlled by the value of the parameter $r$ since $r$ controls the degree of asymmetry of each traveling state, but it also depends on the length of the structure, cf. Fig. 4. For example, in scenario A at $r=-0.7$ a narrow LS catches up to a wider LS, with the resulting interaction stopping the former and leaving the latter unaffected. Because of periodic boundary conditions the wider, slower moving LS collides with the stationary state in a subsequent collision, creating a drifting bound state (not shown, but see Fig. 8 for a similar multiple collision at $r=-0.635$). Such drifting bound states are not observed in any of the other scenarios at this value of $r$ where a narrower and so faster LS interacts with a broader LS at rest. Figure 9 shows an extreme case of scenario B, where a single-wavelength asymmetric LS collides with a stationary symmetric structure and is annihilated. This is because this asymmetric LS is located on the lowest Z branch in Fig. 3 and so is minimal in the sense that no stable LS shorter than one significant wavelength exists. Hence, the deletion of an extremum implies annihilation of this LS.

To quantify the change in pattern size during a collision, we measure the $L_{2}$ norm defined in Eq. (3). Figure 10 shows a sample time series of $\|u\|_{2}$ from a chasing collision (scenario A) at $r=-0.65$, where the collision first produces a metastable bound state, but three new extrema are generated at late times by nonlinear interactions. Figure 10 suggests the simple metric

where $u_{\textrm{initial}}$ is the state before the collision, i.e., when the distance between the two structures is still large, and $u_{\textrm{final}}$ is the resulting state a long time after the collision has occurred. Importantly, the collision dynamics are independent of the initial distance between the colliding LSs, since there is no inertia in the system. This is illustrated in Fig. 11, where the initial distance in scenario B (symmetric-asymmetric collision) at $r=-0.65$ is varied by fractions of the wavelength associated with the spatial eigenvalue, $\lambda=2\pi/\beta$, with $\beta$ defined in Eq. (12). Figure 11 shows that the collision dynamics are self-similar under shifts in the initial distance: simply accounting for the time required to propagate over the additional distance leads to data collapse. This has also been verified explicitly for other collisions, including from scenario D. We deduce from these observations that the only variables affecting collision outcomes are indeed the chosen pair of colliding structures, and the control parameter $r$ (at fixed $\epsilon$).

The phenomenology illustrated in Fig. 10 is reminiscent of collision dynamics described in terms of scattors [24, 25]: unstable stationary or time-periodic patterns which direct the evolution in state space during the collision process along their stable and unstable manifolds. While these ideas were proposed in the context of models other than SH35, they may be applicable for some of the collision events observed here.

Figure 12 shows $\Delta\|u\|_{2}$ as a function of $r$ for all four scenarios. A value of $\Delta\|u\|_{2}\approx 0$ corresponds to the formation of a bound state without a change in the overall pattern size, a positive value indicates the creation of one or more extrema, while a negative value indicates that one or more extrema were deleted.

The range of $r$ shown in Fig. 12 corresponds to the interval with stable propagating solutions when $\epsilon=0.03$. Vertical gray dashed lines indicate the cases depicted in Fig. 7. Near the leftmost edge of the existence interval, at $r=-0.7$, one (A-C) or two extrema (D) are deleted in the collision. Near the rightmost edge, at $r=-0.635$, new extrema are created: either one (A, B) or five extrema (C, D). Away from these boundary cases, the scenarios differ more substantially. At $r=-0.68$, one observes either the formation of bound states (A, B), or the creation of three additional extrema (C, D). At $r=-0.66$, new extrema are added in the collision in all cases: either four additional extrema (A, B) or three (C, D). At $r=-0.645$ and $r=-0.65$ (not shown), scenarios A and C revert to $\Delta\|u\|_{2}\approx 0$, i.e. the formation of a bound state, while new extrema continue to be added in scenarios B (four extrema) and D (three extrema).

As already mentioned in the discussion of Fig. 7, scenarios A and B are similar to one another, as are C and D, in the sense that for most values of $r$, they display the same number of extrema added or deleted. However, Fig. 12 reveals deviations between these scenarios in the interval $-0.65\lesssim r\lesssim-0.64$. To ensure that these are not numerical artefacts, we repeated the runs in scenarios A and B in this range of $r$ at higher spatial resolution, using $8192$ instead of $2048$ grid points, and continued the simulation until very late times ($t\approx 150~000$). We also repeated the runs with significantly smaller time steps, $dt=0.005$ and $dt=0.001$, with the same collision outcome, confirming that the nonmonotonic dependence of $\Delta\|u\|_{2}$ on $r$ is a robust result that we discuss further below.

The black dash-dotted vertical line in Fig. 12 indicates the location of the Maxwell point $r=r_{M}$ for the variational case $\epsilon=0$. The figure shows that the values of $r$ where $\Delta\|u\|_{2}$ changes differ between scenarios and that, in addition, these locations lie far from $r=r_{M}$. We conclude that the Maxwell point of the variational case has little, if any, relevance in determining the collision outcome in the nonvariational case, even in the weak symmetry-breaking case considered here.

The overall trend of increasing $\Delta\|u\|_{2}$ with increasing $r$ can be viewed as a reflection, in the nonvariational regime, of similar behavior in the variational case: when $\epsilon=0$, the free energy $\mathcal{F}$ of the stable pattern state decreases with increasing $r$, as shown in Fig. 2, and the periodic pattern becomes increasingly energetically favored. Apart from some exceptions at $r\geq-0.65$ (to be discussed further below), the increasing trend in the number of extrema added in a collision is compatible with this intuition. Finally, even for a fixed number of extrema added or lost, Fig. 12 reveals a slow decrease in $\Delta\|u\|_{2}$ with increasing $r$. This is associated with small changes in the profile of the extrema as $r$ changes.

Figure-eight isola structures extending over the entire width of the snaking region have been previously described in the quadratic-cubic Swift-Hohenberg equation without nonvariational terms, arising from multi-pulse solutions consisting of two LSs bound together in close proximity [26, 27]. Here, we find that for bound states formed from collisions at $\epsilon>0$, the properties of the isolas to which these states belong depend strongly on the collision scenario.

Figure 13 shows two isolas at $\epsilon=0.03$ (thin lines) obtained from a continuation in $r$, starting from two bound states generated by a collision in scenarios A and B at $r=-0.68$ and $r=-0.69$, respectively (points highlighted in Fig. 13), together with the results of continuing these isolas in $\epsilon$ to the variational case $\epsilon=0$ (thick lines). The figure shows that in the former case (red curves) the isola retains its shape as it transforms into the corresponding isola of the variational problem. In the latter and more typical case, the isola at $\epsilon=0.03$ takes the form of a spaghetti-like tangle (light blue), requiring substantial simplification prior to reaching the corresponding isola of the variational problem (thick blue). Figure 14 shows several additional isolas of traveling bound states at $\epsilon=0.03$ obtained by colliding longer initial LSs and superposed on the full bifurcation diagram (colors match between Figs. 13 and 14). All figure-eight isolas seen in Fig. 14 are from scenario A, while tangled isolas are from scenario B.

It is important to note that the rightmost edge of the figure-eight isola in scenario A shown in Fig. 13 (thin red line) is located near $r\approx-0.66$, and not at $r\approx-0.63$, the right boundary of the snaking region for $\epsilon=0.03$. This is a consequence of the fact that two-pulse states with a small separation between the LSs lie on isolas that are significantly smaller than those for bound states with a larger separation. This is so not only when $\epsilon=0$ [26] but also when $\epsilon=0.03$. Figure 15 shows an example of a narrow figure-eight isola at $\epsilon=0$ obtained from a bound state at $\epsilon=0.03$ and $r=-0.68$ together with a wider figure-eight isola obtained by continuing a bound state, also obtained from a collision in scenario A with $\epsilon=0.03$, but at $r=-0.645$. Despite their similarity (see profiles in Fig. 15, top panel) these profiles are part of separate isolas: the solution profiles of the narrow isola feature an interaction region between the two LSs comprising the bound state which is one wavelength shorter than that of the larger isola solution profiles, a property that is preserved upon numerical continuation. As the separation between the two LSs decreases and their interaction becomes stronger, the figure-eight isolas shrink, with the rightmost edge moving farther and farther to the left.

This decrease in size plays a key role in determining collision outcomes: the value $r\approx-0.66$ (Fig. 13, thin read line) coincides with the parameter value in Fig. 12 where new extrema are first created in scenario A. This observation suggests the following way of understanding the observed phenomenology: when two LSs approach one another in a collision, there are two possibilities. Either (a) a stable multipulse bound state exists at the value of $r$ in question for the given pair of LSs, or (b) no such state exists or at least it is not stable. In case (a), the collision will lead to the formation of a stable bound state. In case (b), on the other hand, no such state is available, and the system creates or deletes extrema to reach another stable solution.

With this hypothesis, the nonmonotonic behavior in the number of extrema created summarized in Fig. 12 can be explained as follows: at parameter values $-0.65\lesssim r\lesssim-0.64$, there is an island of stability where stable bound states exist. When no stable bound state exists, a strongly nonlinear interaction must occur, resulting in LSs with a different number of extrema; this number increases with increasing $r$, as discussed earlier.

To test the hypothesis that the existence and stability of a bound state is the determining factor in setting the collision outcome, we perform stability experiments. Specifically, to verify that the bound state either does not exist or is unstable in scenarios A, B with $-0.66\lesssim r\lesssim 0.65$ (where new extrema form), we performed DNS at these values of $r$, initializing with the post-collision final states from $r=-0.68$, $r=-0.67$, $r=-0.65$ and $r=-0.645$. In each of these cases we observed the eventual generation of additional extrema. Only one exception was observed, at $r=-0.66$ in scenario B, when a bound state from $r=-0.68$ was used as the initial condition and found to remain a pure bound state indefinitely, without any change in the number of extrema. Thus collisions may trigger the creation of new extrema despite the existence of a stable bound state, provided the bound state has only a small basin of attraction.

We repeated the above study for scenario C: stable bound states formed from collisions at $r=-0.645$ and $r=-0.65$ were used as initial conditions at $r=-0.635$, $r=-0.64$, $r=-0.655$, $r=-0.66$. In all cases but one, the same number of extrema was spontaneously created as observed in the collision experiments. The exception, where a bound state remains indefinitely stable, was again found near the transition from a bound state to the creation of new extrema in the collision, namely at $r=-0.64$. In summary, these stability experiments largely confirm the proposed hypothesis for explaining Fig. 12: collisions yield bound states when these exist as stable states, and otherwise lead to a change in the number of extrema (typically creation of new extrema). The only deviations from this paradigm are observed when stable bound states exist but have a small basin of attraction. Then the finite perturbation provided by a collision can trigger a change in the number of extrema.

The isolas corresponding to multi-pulse bound states at $\epsilon>0$ can take other forms as well. Figures 16 and 17 show two such cases, in which a complex tangled branch terminates at either end in bifurcation points located at the center of a spiral structure. These figures correspond to the continuation of bound states resulting from scenario A collisions at $r=-0.645$ and $r=-0.7$, respectively. As mentioned in Sec. IV.1, the latter is in fact the result of two separate collisions. The first collision occurs between two moving LSs and leaves one stationary and symmetric LS and one moving LS as shown in Fig. 7. The second collision (not shown) occurs after this, when the moving LS collides with the stationary LS from the other side due to periodic boundary conditions, forming a traveling bound state.

The bifurcations identified in Figs. 16 and 17 are of codimension two and occur as the separation between the two LSs in the bound state grows without limit, while the structures independently change their amplitude and degree of asymmetry such that their velocities remain the same. As the branch spirals towards the center, the separation between the constituent LSs grows. These bifurcations are highly reminiscent of T-point bifurcations, also known as Bykov points [28, 29], which have been described as corresponding to an equilibrium-to-equilibrium heteroclinic cycle, with two branches of primary homoclinic orbits bifurcating from it. Bifurcations of this type have been observed in a variety of systems, including the Lorenz63 system [30], a model of calcium pulses in pancreatic cells [31], and in type-I excitable media [32]. In contrast, here the two constituent LSs individually approximate a homoclinic orbit to the trivial state $u=0$ and at the T-point the state of the system corresponds to a double homoclinic cycle. Of course, in a finite domain we cannot reach this T-point, and our numerical continuation results therefore only produce a periodic structure that approximates this double homoclinic cycle. To the best of our knowledge, a T-point bifurcation from such a double homoclinic cycle described here has not been observed previously. Note that such T-points cannot occur for stationary LSs.

It is interesting to note that in all the cases described above, when the interacting extrema were of opposite signs (scenarios A, B), we found $g_{1}>0$ and $g_{2}<0$ (for $-\pi<\phi<0$). Conversely, when the two interacting extrema were of the same sign (scenarios C, D), we systematically found $g_{1}<0$ and $g_{2}>0$ (for $-\pi<\phi<0$). However, it is nontrivial to deduce what this implies for the sign of the effective interaction, due to its oscillatory nature. We can define an effective sign of the interaction as follows. First we introduce the equilibrium distance $d_{eq}$ as the value of $|x_{2}-x_{1}|$ in the bound state after the collision has occurred. From Eq. (13), we see that equilibrium implies

In general, Eq. (15) has multiple solutions, as illustrated in Fig. 27. One may surmise that these multiple solutions are related to the overlapping isolas shown in Fig. 15, which differ in their equilibrium distance by one wavelength. When $c_{2}=c_{1}$, Eq. (15) has infinitely many solutions, similar to the family of bound two-pulse states described in [26] that differ only in their equilibrium separation. By contrast, for $c_{2}-c_{1}\neq 0$, the number of solutions is finite. In a generic collision at $\epsilon>0$, since the colliding LSs approach from a large distance, we expect that the physical value of $d_{eq}$ is given by the largest positive value of $d$ that solves Eq. (15).

We assume that $d_{eq}$ is known for a given collision of two LSs starting out at a distance $|x_{2}-x_{1}|=d_{0}$, with $c_{1}>c_{2}$ (which applies to all examples shown above). In the absence of interactions, the free propagation of the two LSs will reduce the distance from $d_{0}$ to $d_{eq}$ in a time

In the presence of interactions, this time will change to $t_{eq}$, a time that may be larger or smaller than $t_{free}$ due to the oscillatory nature of the interaction.

We propose the following terminology. If the equilibrium distance in the collision is reached earlier than for free propagation, i.e. if $t_{eq}<t_{free}$, then we say that the interaction is effectively attractive. By contrast, if the equilibrium distance is reached later, i.e. $t_{eq}>t_{free}$, then we say that the interaction is effectively repulsive.
We note that the effective sign of the interaction can be determined graphically from the sign of $\chi(t_{eq})$, since Eq. (14) implies

The time $t=t_{eq}$, where the distance $|x_{2}-x_{1}|=d_{eq}$ is highlighted by an arrow in Figs. 19, 22, 24 and 26; since $x_{2}-x_{1}$ is constant at $t>t_{eq}$, a linear increase in $\chi=const+(c_{1}-c_{2})t$ sets in. The sign of $\chi(t_{eq})$, i.e. the onset of this linear increase, determines the effective sign: if $\chi(t_{eq})>0$, then the interaction is effectively repulsive but if $\chi(t_{eq})<0$, then the interaction is effectively attractive.

We highlight that even though the collisions from scenarios A and B shown in Figs. 18 and 21 both feature interactions between extrema of opposite signs, the relative signs of their interaction differ: the interaction is effectively repulsive in scenario A (Fig. 19), while it is effectively attractive in scenario B (Fig. 22). This indicates that while the relative sign of interacting extrema does appear to be important, it is not the only factor determining collision outcomes. This is likely related to the observed deviation between scenarios A, B and C, D, respectively, in the parameter range $-0.65\lesssim r\lesssim-0.64$.